D₃₀₇₀₄ #72296017

order	61408
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₀₇₀₄

Splittings

Fitting	C₁₆×C₁₉×C₁₀₁ ⋊ C₂
Hall	C₁₉×C₁₀₁ ⋊ D₁₆
Sylow	C₁₀₁ ⋊ D₃₀₄
Sylow	C₁₉ ⋊ D₁₆₁₆

Subgroups

	subgroup	G/H
center	C₂	D₁₅₃₅₂
commutator, cosocle	C₈×C₁₉×C₁₀₁	C₂²
Frattini	C₈	C₂×D₁₉₁₉
Fitting	C₁₆×C₁₉×C₁₀₁	C₂
socle	C₂×C₁₉×C₁₀₁	D₈
hypercenter	C₁₆	D₁₉₁₉
hypocenter	C₁₉×C₁₀₁	D₁₆
LCS	G ▷ C₈×C₁₉×C₁₀₁ ▷ C₄×C₁₉×C₁₀₁ ▷ C₂×C₁₉×C₁₀₁ ▷ C₁₉×C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 136
transitive degree	μ_t(G) = 30704

A minimal faithful permutation representation — 2 generators on 136 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(37,136)(38,135)(39,134)(40,133)(41,132)(42,131)(43,130)(44,129)(45,128)(46,127)(47,126)(48,125)(49,124)(50,123)(51,122)(52,121)(53,120)(54,119)(55,118)(56,117)(57,116)(58,115)(59,114)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)

Polycyclic presentation:

2^5,19,101|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^18,g^a=g^100

Conjugacy classes & character table

15355 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 19, 101]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^18);;
gap> SetConjugate(coll, 7, 1, F.7^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30704 #72296017